What is the difference between portamento and glissando while singing?
Glissando is a discrete portamento whereas portamento is a continuous glissando.
Although the two terms are often conflated, and although there are plenty of situations where the difference isn't noticeable, the intended difference is that portamento is a relatively quick pitch slide between all or most of the notes during a melody or melody section. The pitch slide tends to come only at the very end of the note, more or less during the articulation change into the next pitch. A glissando is a far more deliberate slide that generally lasts for a significant part of the duration of the initial pitch on its way to the new pitch.
In short, portamento is a relatively subtle effect that tends to apply to the overall singing style while glissando tends to be a more dramatic pitch shifting that is usually only done between two notes where it is explicitly notated.
Irrespective of singing, a glissando indicates that specific notes (whether specified or not) should be played or performed. For example, if there is a perfect-fifth leap with a port. indication, the performer should play/sing all of the chromatic notes in between unless otherwise indicated.
A portamento indicates that no specific pitches are necessary and may include semi or quarter tones as well. A glissando is more gestural and is often larger than a portamento.
In singing, Portamento is a style, Glissando a technique. Neither are likely to be heard as a distinct chromatic (or otherwise) scale.
On a harp, Glissando is a scale. On trombone it's a continuous smear.
The terms mean different things in different contexts.
Glissando is going through intermediate pitches musically. Portamento "carries" one pitch to another pitch: there is no interruption in tone and style and no "musical concept" of intermediate notes even though the execution might not be able to switch pitches instantly. It's pretty much the same as a slur over a larger interval. If you map this to a violin, it would a slur across two notes on the same string, played with the same finger but shifting position.
Monti's "Czárdás" starts with a violin passage all on one string, with large legato lines. There is no word of "glissando" in the actual score if memory serves me right: the instruction was only "sul G". Various interpretations show a wide variety of executions for the large jumps, some are really exuberant glissandi, some are actually more like portamento. The latter is quite trickier to pull off well and smoothly.
The basic difference is that there is no "musical time" spent on a portamento.
When using "Czardas" as an example for portamento identity, it's important to note that the interpretive portion of the piece is heavily rubato, disregarding time/tempo and leaving much to expression. A clear definition of portamento (as I perform it, attacking a note and sliding to another with no discernable voicings in between,and almost immediately following a stroke) will add that much more to your expression.